October 16, 2013: Koichi Kaizuka (University of Tsukuba), A characterization
of the L2 -range of the Poisson transform on symmetric spaces of noncompact type.
Characterizations of the joint eigenspaces of invariant differential operators have
been one of the central problems in harmonic analysis on symmetric spaces. In
1970 Helgason conjectured that any joint eigenfunction on symmetric spaces of
noncompact type is expressed as the image of the Poisson transform of an analytic functional on the boundary, and this conjecture was proved by six Japanese
mathematicians in 1978.
In 1989, from the point of view of spectral theory, Strichartz formulated a conjecture concerning a different image characterization of the Poisson transform of the
L2 -space on the boundary. Except for the rank one case, the Strichartz conjecture
had remain unsolved up to the present moment.
In this talk, we employ techniques in scattering theory to present a positive
answer to the Strichartz conjecture in the general case. By virtue of the Fourier
restriction estimate for the Helgason Fourier transform and the scattering formula
for the Poisson transform, we give global estimates for joint eigenfunctions in terms
of the Agmon-Hörmander type norms.